Abstracts

September 9: Shuai Wang

Title: Non-recursive formula for Kazhdan-Lusztig polynomials for finite and affine G/P

Abstract:

Brenti proves a non-recursive formula for the Kazhdan-Lusztig polynomials of Coxeter groups by combinatorial methods. In the case of the Weyl group of a split group over a finite field, a geometric interpretation of the formula is given by Sophie Morel via the "weight-truncation" theory of perverse sheaves developed by her. With suitable modifications of Morel's proof, we generalize the geometric interpretation to the case of finite and affine partial flag varieties. We also demonstrate the result for sl_3, sl_4 and affine sl_2.

September 16 and 23: Ivan Danilenko

Title: Slices of the Affine Grassmannian and Quantum Cohomology

Abstract:

The Affine Grassmannian is an ind-scheme associated to a reductive group $G$. It has a cell structure similar to the one in the usual Grassmannian. Transversal slices to these cells give an interesting family of Poisson varieties. Some of them admit a smooth symplectic resolution and have an interesting geometry related to the representation theory of the Langlands dual group. We will focus on equivariant cohomology of such resolutions and will show how the Knizhnik-Zamolodchikov equation arises as a quantum difference equation in this setting.

October 7: Marco Castronovo

Title: Lagrangian tori and cluster charts

Abstract:

I will describe a conjectural correspondence between Lagrangian tori in a real symplectic manifold and algebraic tori in a mirror variety. It is not clear what this mirror should be, but for coadjoint orbits work of Rietsch suggests a relation to Langlands duality. I will then explain how to partially verify this correspondence for Grassmannians. This point of view allows to answer purely dynamical questions about displaceability and abundance of Lagrangians. I will end with speculations on how symplectic topology might one day return the favor.

October 14: Amol Aggarwal

Title: Universality for Lozenge Tiling Local Statistics

Abstract:

We consider uniformly random lozenge tilings of essentially arbitrary domains and show that the local statistics of this model around any point in the liquid region of its limit shape are given by the infinite-volume, translation-invariant, extremal Gibbs measure of the appropriate slope. In this talk, we outline a proof of this result, which proceeds by locally coupling a uniformly random lozenge tiling with a model of Bernoulli random walks conditioned to never intersect. Central to implementing this procedure is to establish a local law for the random tiling, which states that the associated height function is approximately linear on any mesoscopic scale.

October 21 (3:00): Philippe Di Francesco

Title: Dualities in difference Toda equations: from Macdonald theory to quantum cluster algebra

Note the unusual time & location !

October 21 (5:40) : Richard Rimanyi

Title: Characteristic classes of singular varieties in H, K, and Ell

Abstract: Characteristic classes associated with singular varieties are powerful tools to study the topology and geometry of the singularities. We will review the foundations of the theory, explore calculational techniques (resolution, degeneration, localization, interpolation), deduce enumerative results, and we will present the connection between characteristic classes and stable envelopes.

October 28 (11:00): Alexei Oblomkov

Title: Moduli space of quilts and relative Fulton-MacPherson spaces

Abstract:

Talk is based on the joint work with Nate Bottman. Motivated by the quilt theory in symplectic geometry we introduce some natural compactification of the moduli space of pointed vertical lines. The topological structure of this is determined by the Gromov convergence, we provide this space with the structure of an algebraic variety and show this space has at most toric normal lci singularities. We also give an interpretation of these spaces in terms of relative Fulton-MacPherson spaces.

In my talk I will concentrate on the simplified version of or construction that provides us with an explicit atlas on the space of stable pointed rational curves.

October 28 (5:40): Junliang Shen

Title: Hitchin systems, hyper-Kaehler geometry, and the P=W conjecture

Abstract:

Lagrangian fibrations play a crucial role in the study of hyper-Kaehler geometry and integrable systems. The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Lagrangian fibrations and Hodge theory. In this talk, we will first discuss a compact version of this phenomenon, based on joint work with Andrew Harder, Zhiyuan Li, and Qizheng Yin. Then we will focus on interactions between compact and noncompact hyper-Kaehler geometry. Such connections lead to new progress on the P=W conjecture for Hitchin systems and character varieties. This is joint work with Mark de Cataldo and Davesh Maulik.

November 11: Guillaume Remy

Title: A probabilistic approach to conformal blocks and conformal bootstrap for Liouville theory

Abstract:

Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by A. Polyakov in the context of string theory. As for all CFT it is in principle exactly solvable using the conformal bootstrap formalism. Very recently Liouville theory has been rigorously constructed using a probabilistic framework. In this talk we will present work in progress to understand the conformal blocks and the conformal bootstrap for Liouville theory using probability. We will also mention some connections with the AGT correspondence. Based on joint work with Promit Ghosal, Xin Sun, and Yi Sun.

November 18: Ben Wormleighton

Title: McKay correspondence and walls for G-Hilb

Abstract:

The McKay correspondence takes many guises but at its core connects the geometry of minimal resolutions for quotient singularities C^n / G to the representation theory of the group G. I will introduce the classical situation for SL(2), along with its categorification and extension to three dimensions. When G is an abelian subgroup of SL(3), Craw-Ishii showed that every minimal resolution can be realised as a moduli space of stable quiver representations, although the chamber structure for the stability parameter and associated wall-crossing behaviour is in general poorly understood. I will describe my recent work computing the walls and wall-crossing behaviour for the chamber corresponding to a particular minimal resolution called the G-Hilbert scheme. Time permitting, I will also discuss ongoing work with Yukari Ito (IPMU) and Tom Ducat (Bristol) to better understand the geometry, chambers, and corresponding representation theory for other minimal resolutions.

November 25: Justin Hilburn

Title: 3d Mirror Symmetry and (S-)ymplectic duality

Abstract:

Thanks to work of Kapustin-Witten it is well known that geometric Langlands is the mathematical incarnation of S-duality of topologically twisted 4d N=4 gauge theories. It is less well known that that the mathematical theory of symplectic duality is a manifestation of mirror symmetry for 3d N=4 supersymmetric theories. (Not to be confused with what the mathematical theory of mirror symmetry which is mirror symmetry for 2d N=(2,2) theories.) In this talk I will show how understanding the relationship between these physical theories lets one deduce interesting mathematical conjectures. Much of this work is joint with various subsets of Tudor Dimofte, Philsang Yoo, Joel Kamnitzer, Alex Weekes, and Nik Garner with lots of help from Davide Gaiotto, Kevin Costello, and Ben Webster.

December 9: Ben Gammage

Title: Hyperkahler geometry and (2d) mirror symmetry

Abstract:

Focusing on the particular example of multiplicative hypertoric varieties, we discuss the symplectic geometry of spaces which appear as moduli of 4d N=2 theories. We will explain both structural features of this geometry and some concrete calculations which can be used to prove homological mirror symmetry. This is based on joint work with Michael McBreen and Ben Webster, and possibly also on work with Ian Le.

January 27: Gus Schrader

Title: Decorated character varieties and their cluster structure via factorization homology

Abstract:

Character varieties for surfaces are affine, generally singular, Poisson varieties, for which quantizations were constructed by Fock-Rosly and Alekseev-Grosse-Schomerus in the 90's. Subsequently Fock and Goncharov introduced the notion of a decorated character variety corresponding to a marked surface, a rational Poisson variety with a Poisson projection to the original character variety of the unmarked surface. Moreover, the ring of functions on a decorated character variety carries the additional structure of a cluster algebra. In general, however, the decorated character variety is not affine, and quantizing it amounts to constructing a quantum analog of its category of quasicoherent sheaves. I'll explain an approach to constructing such a quantization using the stratified factorization homology developed by Ayala-Francis-Tanaka, and show how one can recover the combinatorics of cluster charts on the decorated character variety from this construction. Joint work with D. Jordan, I. Le and A. Shapiro.

May 11: Song Yu

Title: The Open Crepant Transformation Conjecture for toric Calabi-Yau 3-orbifolds

Abstract:

I will discuss an open version of Ruan’s Crepant Transformation Conjecture, which is an identification of all-genus open-closed Gromov-Witten invariants of K-equivalent toric Calabi-Yau 3-orbifolds. Our approach is based on the mirror symmetry between toric Calabi-Yau 3-orbifolds and B-model mirror curves. I will first discuss the case of disk invariants, proven by the construction of a global family of mirror curves over the B-model moduli space and the disk mirror theorem of Fang-Liu-Tseng. I will then discuss ongoing joint work with B. Fang, C.-C. Liu, and Z. Zong on the general case based on the BKMP Remodeling Conjecture.

May 18: Konstantin Aleshkin

Title: Liouville quantum gravity and integrable systems

Abstract:

Correlation numbers of 2d Liouville quantum gravity are defined as integrals of certain products of conformal blocks over moduli spaces of punctured curves and are quite challenging to compute directly. The other major approaches to 2d quantum gravity: topological gravity, and matrix models are much better understood. The connection between the topological gravity and matrix models is established by the Witten conjecture and its generalizations. In the talk I will speculate on the connection of the Liouville gravity and the other two approaches.

May 25: Renata Picciotto

Title: Stable maps with p-fields

Abstract:

The moduli space of stable maps with p-fields was first introduced by Huai-liang Chang and Jun Li to study the higher genus Gromov-Witten invariants of the quintic threefold, and has since seen various generalizations. The goal is to study stable maps to a target Z which is not toric and does not naturally lend itself to virtual localization techniques. If Z can be realized as a vanishing locus in some space X, stable maps to Z can be related to maps with p-fields to X. In this talk, I will discuss the moduli of stable maps with p-fields on a smooth projective variety X.

June 1: Sam DeHority

Title: Geometric Lie Algebra Actions on Moduli Spaces for K3 Surfaces

Abstract:

It is well known that various Lie algebras act on the cohomologies of moduli spaces of sheaves on surfaces. The situation is best understood when the surface is an ADE surface, and Lagrangian correspondences between Nakajima quiver varieties give representations of affine Lie algebras on birational models of moduli spaces of torsion free sheaves on the surface. It is possible to extend some of these results to the case of some K3 surfaces which have -2 curves arranged according to a Lorentzian root system and this is related to the birational geometry of the Hilbert scheme of points on K3 surfaces provided by variation of Bridgeland stability conditions.

June 8: Yakov Kononov

Title: Relative M-theory

Abstract:

The correspondence between PT-theory and M-theory was discovered in 2014 by N.Nekrasov and A.Okounkov. I will talk about imposing boundary conditions on the membranes which allows to prove the conjecture by degeneration of the base curve.

June 15: Anton Mellit

Title: Counting bundles and Macdonald polynomials

Abstract:

Many interesting functions arise as generating functions of geometric invariants. There are different kinds of invariants one can consider, such as Euler characteristics of spaces or holomorphic vector bundles or their equivariant generalizations. I will talk about another way to produce invariants: counting points over finite fields. I will explain how the Hall-Littlewood polynomials (both modified and unmodified) can be obtained this way. Then I will explain how to obtainMacdonald polynomials and generating functions considered by Hausel, Letellier and Rodriguez-Villegas (a type of Nekrasov partition functions) by counting bundles with endomorphisms or twisted endomorphisms.

June 22: Henry Liu

Title: Quasimaps and stable pairs

Abstract:

I will give a short introduction to the various flavors of moduli of 1-dimensional sheaves on threefolds (e.g. Donaldson-Thomas theory), all related by wall-crossing between certain stability chambers in the derived category. One such chamber, first studied by Bryan and Steinberg, yields the theory of pi-stable pairs. I will explain why pi-stable pairs and quasimaps are equivalent whenever they are comparable. Quasimaps have been used recently to study 3d mirror symmetry, which when pushed through this equivalence has implications for some aspects of sheaf-counting theories, including the (DT) crepant resolution conjecture. If time permits I'll discuss the proof of the equivalence, which explicitly matches vertices for the two theories using the derived McKay equivalence.

June 29: Roman Gonin (Moscow, HSE&Skoltech)

Title: Twisted representations of toroidal gl_1

Abstract:

Fock module is a basic representation of quantum toroidal gl1; it can be identified with equivariant K-theory of Hilbert scheme of points on C^2. We study a twisted Fock module which is the same vector space with an action "twisted by a certain automorphism of the algebra". Surprisingly, an attempt "to make this action explicit" leads to an appearance of an auxiliary quantum affine gl_n-action on (twisted) Fock space. I will explain our purely algebraic construction and formulate a conjectural application to geometry (conjecture of Gorsky and Negut on K-theoretic stable bases).

July 6: Noah Arbesfeld

Title: Boxcounting and Quot schemes

Abstract:

I'll present concise expressions for the generating series of virtual holomorphic Euler characteristics parametrizing 0-dimensional quotients on C^2 and C^3, along with some generalizations. The formulas in the surface case yield information about certain Nekrasov partition functions, while those for threefolds are related to a conjectural framework of Nekrasov and Okounkov comparing the DT theory of threefolds with the geometry of certain Calabi-Yau fivefolds. Most of what I say will be joint work with Yakov Kononov. I'll also mention some results from work with D. Johnson, W. Lim, D. Oprea and R. Pandharipande.

Minicourse starting July 13 at 3:00 (note special time!):

Lev Rozansky

2-categories of affine Nakajima quivers, categorical representations of braid groups and a categorification of HOMFLY-PT polynomial

Abstract:

This is a joint work with Alexei Oblomkov. I will try to explain the following:

1. To a symplectic variety X one associates a special 2-category B(X). Its objects are fibrations whose bases are Lagrangian subvarieties of X. The morphisms between objects are defined as categories of matrix factorizations. Generally, this construction is work in progress. When X is a cotangent bundle: X = T*Y, the 2-category B(X) is (presumably) equivalent to the 2-category of Y appearing in `derived algebraic geometry'.

2. The definition of B(X) is rigorous when X is (an open subvariety of) a cotangent bundle with the Hamiltonian action of a group G. In particular, B(X) can be defined for Nakajima quiver (and Nakajima-Cherkis quiver-bow) varieties.

3. Consider a special case when X is the variety of an affine quiver corresponding to the moduli space of n U(r) instantons on C^2. For a special object FL in B(X) (related to a flag variety) we define a homomorphism from the (affine) braid group on n strands to the monoidal category End(FL). When r=0 (`commuting variety') it categorifies the affine Hecke algebra (equivalent to Bezrukavnikov-Riche), when r=1 (Hilbert scheme of points on C^2) it categorifies the ordinary Hecke algebra (equivalent to Soergel-Rouquier) and for r>1 it categorifies the cyclotomic Hecke algebra.

4. In case of r=1 the space of morphisms between the images of a braid and the identity braid categorifies the HOMFLY-PT polynomial of the braid closure. Then the categorical version of the Riemann-Roch theorem says that to a closed braid one associates a complex of sheaves on the Hilbert scheme whose homology coincides with the link homology.

5. The object FL comes from the `quiver-in-quiver' construction related to a 3d TQFT with defects. In Nakajima quiver terms, we interpret a circle with number n as 2-category of a moduli space of n instantons and an edge connecting two such circles as a functor of a Lagrangian correspondence between two instanton moduli spaces. Then an A-type quiver 0-1-2- … - n - … - 2-1-0 becomes the monoidal category End(FL) used for braid representations. From the 3d TQFT perspective, we consider a surface split into regions by intersecting curves: the regions are `colored’ with instanton moduli spaces, the curves are colored by Nakajima-type Lagrangian correspondences and the intersection points are colored by matrix factorizations corresponding to braid crossings.